In this paper we propose a novel distributed state estimator for large-scale linear systems composed by subsystems interacting through state variables. The distributed state estimator has the following features: (i) local state estimators, each dedicated to the reconstruction of the states of a subsystem, are connected through a communication network with the parent-child topology induced by subsystems coupling; (ii) the design of a local state estimator requires information on the associated subsystem and its parents only. As a consequence, both the offline design and the online implementation are distributed and scalable. In particular, the addition and removal of subsystems can be handled in a plug-and-play fashion. The distributed state estimator is also combined with a plug-and-play distributed model predictive control scheme to provide a novel output-feedback plug-and-play distributed controller capable of guaranteeing nominal convergence and constraint satisfaction. Applications to a mechanical system and power networks demonstrate the effectiveness of the approach.